model JointUPS "Universal - prismatic - spherical joint aggregation (no constraints, no potential states)"
import Modelica.Mechanics.MultiBody.Types;
extends Interfaces.PartialTwoFramesDoubleSize;
Modelica.Mechanics.MultiBody.Interfaces.Frame_a frame_ia "Coordinate system at origin of frame_a fixed at prismatic joint"
annotation (Placement(transformation(
origin = {-80, 100},
extent = {
{-8, -8},
{8, 8}},
rotation = 270)));
Modelica.Mechanics.MultiBody.Interfaces.Frame_b frame_ib "Coordinate system at origin of frame_b fixed at prismatic joint"
annotation (Placement(transformation(
origin = {80, 100},
extent = {
{-8, 8},
{8, -8}},
rotation = 270)));
Modelica.Mechanics.Translational.Interfaces.Flange_a axis "1-dim. translational flange that drives the prismatic joint"
annotation (Placement(transformation(extent = {
{45, 95},
{35, 105}})));
Modelica.Mechanics.Translational.Interfaces.Flange_b bearing "1-dim. translational flange of the drive bearing of the prismatic joint"
annotation (Placement(transformation(extent = {
{-35, 95},
{-45, 105}})));
parameter Boolean animation = true "= true, if animation shall be enabled";
parameter Boolean showUniversalAxes = true "= true, if universal joint shall be visualized with two cylinders, otherwise with a sphere (provided animation=true)";
parameter Modelica.Mechanics.MultiBody.Types.Axis n1_a = {0, 0, 1} "Axis 1 of universal joint resolved in frame_a (axis 2 is orthogonal to axis 1 and to line from universal to spherical joint)"
annotation (Evaluate = true);
parameter SI.Position nAxis_ia[3] = {1, 0, 0} "Axis vector along line from origin of frame_a to origin of frame_b, resolved in frame_ia"
annotation (Evaluate = true);
parameter SI.Position s_offset = 0 "Relative distance offset (distance between frame_a and frame_b = s(t) + s_offset)";
parameter SI.Diameter sphereDiameter = world.defaultJointLength "Diameter of spheres representing the spherical joints"
annotation (Dialog(
tab = "Animation",
group = "if animation = true",
enable = animation));
input Types.Color sphereColor = Modelica.Mechanics.MultiBody.Types.Defaults.JointColor "Color of spheres representing the spherical joints"
annotation (Dialog(
colorSelector = true,
tab = "Animation",
group = "if animation = true",
enable = animation));
parameter SI.Diameter axisDiameter = sphereDiameter / Types.Defaults.JointRodDiameterFraction "Diameter of cylinder on the connecting line from frame_a to frame_b"
annotation (Dialog(
tab = "Animation",
group = "if animation = true",
enable = animation));
input Types.Color axisColor = Modelica.Mechanics.MultiBody.Types.Defaults.SensorColor "Color of cylinder on the connecting line from frame_a to frame_b"
annotation (Dialog(
colorSelector = true,
tab = "Animation",
group = "if animation = true",
enable = animation));
input Types.SpecularCoefficient specularCoefficient = world.defaultSpecularCoefficient "Reflection of ambient light (= 0: light is completely absorbed)"
annotation (Dialog(
tab = "Animation",
group = "if animation = true",
enable = animation));
parameter SI.Distance cylinderLength = world.defaultJointLength "Length of cylinders representing the two universal joint axes"
annotation (Dialog(
tab = "Animation",
group = "if animation = true and showUniversalAxes",
enable = animation and showUniversalAxes));
parameter SI.Distance cylinderDiameter = world.defaultJointWidth "Diameter of cylinders representing the two universal joint axes"
annotation (Dialog(
tab = "Animation",
group = "if animation = true and showUniversalAxes",
enable = animation and showUniversalAxes));
input Types.Color cylinderColor = Modelica.Mechanics.MultiBody.Types.Defaults.JointColor "Color of cylinders representing the two universal joint axes"
annotation (Dialog(
colorSelector = true,
tab = "Animation",
group = "if animation = true and showUniversalAxes",
enable = animation and showUniversalAxes));
parameter Boolean checkTotalPower = false "= true, if total power flowing into this component shall be determined (must be zero)"
annotation (Dialog(tab = "Advanced"));
final parameter Real eAxis_ia[3](each final unit = "1") = Modelica.Math.Vectors.normalizeWithAssert(nAxis_ia) "Unit vector from origin of frame_a to origin of frame_b, resolved in frame_ia";
final parameter Real e2_ia[3](each final unit = "1") = Modelica.Math.Vectors.normalizeWithAssert(cross(n1_a, eAxis_ia)) "Unit vector in direction of second rotation axis of universal joint, resolved in frame_ia";
final parameter Real e3_ia[3](each final unit = "1") = cross(eAxis_ia, e2_ia) "Unit vector perpendicular to eAxis_ia and e2_ia, resolved in frame_ia";
SI.Position s "Relative distance between frame_a and frame_b along axis nAxis = s + s_offset";
SI.Force f "= axis.f (driving force in the axis; = -bearing.f)";
SI.Length axisLength "Distance between frame_a and frame_b";
SI.Power totalPower "Total power flowing into this element, if checkTotalPower=true (otherwise dummy)";
protected
SI.Force f_c_a[3] "frame_ia.f resolved in frame_a";
SI.Torque t_cd_a[3] "frame_ia.t + frame_ib.t resolved in frame_a";
SI.Force f_bd_a[3] "frame_b.f + frame_ib.f resolved in frame_a";
SI.Position rAxis_0[3] "Position vector from origin of frame_a to origin of frame_b resolved in world frame";
SI.Position rAxis_a[3] "Position vector from origin of frame_a to origin of frame_b resolved in frame_a";
Real eAxis_a[3](each final unit = "1") "Unit vector in direction of rAxis_a, resolved in frame_a";
Real e2_a[3](each final unit = "1") "Unit vector in direction of second rotation axis of universal joint, resolved in frame_a";
Real e3_a[3](each final unit = "1") "Unit vector perpendicular to eAxis_a and e2_a, resolved in frame_a";
Real n2_a[3](each final unit = "1") "Vector in direction of second rotation axis of universal joint, resolved in frame_a";
Real length2_n2_a(unit = "1") "Square of length of vector n2_a";
Real length_n2_a(unit = "1") "Length of vector n2_a";
Real der_rAxis_a_L[3](each unit = "1/s") "= der(rAxis_a)/axisLength";
SI.AngularVelocity w_rel_ia1[3];
Frames.Orientation R_ia1_a;
Frames.Orientation R_ia2_a;
Frames.Orientation R_ia_a "Rotation from frame_a to frame_ia";
Visualizers.Advanced.Shape axisCylinder(shapeType = "cylinder", color = axisColor, specularCoefficient = specularCoefficient, length = axisLength, width = axisDiameter, height = axisDiameter, lengthDirection = eAxis_ia, widthDirection = e2_ia, r = frame_ia.r_0, R = frame_ia.R) if world.enableAnimation and animation;
Visualizers.Advanced.Shape sphericalShape_b(shapeType = "sphere", color = sphereColor, specularCoefficient = specularCoefficient, length = sphereDiameter, width = sphereDiameter, height = sphereDiameter, lengthDirection = {1, 0, 0}, widthDirection = {0, 1, 0}, r_shape = {-0.5, 0, 0} * sphereDiameter, r = frame_b.r_0, R = frame_b.R) if world.enableAnimation and animation;
Visualizers.Advanced.Shape sphericalShape_a(shapeType = "sphere", color = sphereColor, specularCoefficient = specularCoefficient, length = sphereDiameter, width = sphereDiameter, height = sphereDiameter, lengthDirection = {1, 0, 0}, widthDirection = {0, 1, 0}, r_shape = {-0.5, 0, 0} * sphereDiameter, r = frame_a.r_0, R = frame_a.R) if world.enableAnimation and animation;
Visualizers.Advanced.Shape universalShape1(shapeType = "cylinder", color = cylinderColor, specularCoefficient = specularCoefficient, length = cylinderLength, width = cylinderDiameter, height = cylinderDiameter, lengthDirection = n1_a, widthDirection = {0, 1, 0}, r_shape = -n1_a * (0.5 * cylinderLength), r = frame_a.r_0, R = frame_a.R) if world.enableAnimation and animation and showUniversalAxes;
Visualizers.Advanced.Shape universalShape2(shapeType = "cylinder", color = cylinderColor, specularCoefficient = specularCoefficient, length = cylinderLength, width = cylinderDiameter, height = cylinderDiameter, lengthDirection = e2_ia, widthDirection = {0, 1, 0}, r_shape = -e2_ia * (0.5 * cylinderLength), r = frame_ia.r_0, R = frame_ia.R) if world.enableAnimation and animation and showUniversalAxes;
equation
if checkTotalPower then
totalPower = frame_a.f * Frames.resolve2(frame_a.R, der(frame_a.r_0)) + frame_b.f * Frames.resolve2(frame_b.R, der(frame_b.r_0)) + frame_ia.f * Frames.resolve2(frame_ia.R, der(frame_ia.r_0)) + frame_ib.f * Frames.resolve2(frame_ib.R, der(frame_ib.r_0)) + frame_a.t * Frames.angularVelocity2(frame_a.R) + frame_b.t * Frames.angularVelocity2(frame_b.R) + frame_ia.t * Frames.angularVelocity2(frame_ia.R) + frame_ib.t * Frames.angularVelocity2(frame_ib.R) + axis.f * der(axis.s) + bearing.f * der(bearing.s);
else
totalPower = 0;
end if;
assert(noEvent(1e-10 < length2_n2_a), "\nA Modelica.Mechanics.MultiBody.Joints.Assemblies.JointUPS joint (consisting of\na universal, prismatic and spherical joint) is in the singular\nconfiguration of the universal joint. This means that axis 1 of\nthe universal joint defined via parameter \"n1_a\" is parallel to vector\n\"eAxis_ia\" that is directed from the origin of frame_a to the\norigin of frame_b. You may try to use another \"n1_a\" vector.\n");
assert(1e-15 < axisLength, "\nDistance between frame_a and frame_b of a JointUPS joint\nbecame zero. This is not allowed. If this occurs during\ninitialization, the initial conditions are probably wrong.");
Connections.branch(frame_a.R, frame_ia.R);
Connections.branch(frame_ia.R, frame_ib.R);
R_ia1_a = Frames.from_T(transpose([eAxis_a,e2_a,e3_a]), w_rel_ia1);
R_ia2_a = Frames.from_T([eAxis_ia,e2_ia,e3_ia], zeros(3));
R_ia_a = Frames.absoluteRotation(R_ia1_a, R_ia2_a);
e2_a = n2_a / length_n2_a;
e3_a = cross(eAxis_a, e2_a);
eAxis_a = rAxis_a / axisLength;
f_bd_a = -eAxis_a * f - e2_a * (n1_a * t_cd_a / (axisLength * (n1_a * e3_a))) + e3_a * (e2_a * t_cd_a / axisLength);
f_c_a = Frames.resolve1(R_ia_a, frame_ia.f);
n2_a = cross(n1_a, eAxis_a);
rAxis_0 = frame_b.r_0 - frame_a.r_0;
rAxis_a = Frames.resolve2(frame_a.R, rAxis_0);
t_cd_a = Frames.resolve1(R_ia_a, frame_ia.t + frame_ib.t);
w_rel_ia1 = {e3_a * cross(n1_a, der_rAxis_a_L) / length_n2_a, -e3_a * der_rAxis_a_L, e2_a * der_rAxis_a_L};
axisLength = s + s_offset;
axisLength = sqrt(rAxis_0 * rAxis_0);
der_rAxis_a_L = (Frames.resolve2(frame_a.R, der(rAxis_0)) - cross(frame_a.R.w, rAxis_a)) / axisLength;
length2_n2_a = n2_a * n2_a;
length_n2_a = sqrt(length2_n2_a);
axis.f = f;
axis.s = s;
bearing.s = 0;
frame_ia.R = Frames.absoluteRotation(frame_a.R, R_ia_a);
frame_ia.r_0 = frame_a.r_0;
frame_ib.R = frame_ia.R;
frame_ib.r_0 = frame_b.r_0;
zeros(3) = frame_a.f + f_c_a + f_bd_a;
zeros(3) = frame_a.t + t_cd_a + cross(rAxis_a, f_bd_a);
zeros(3) = frame_b.t;
zeros(3) = frame_b.f + Frames.resolveRelative(frame_ib.f, frame_ib.R, frame_b.R) - Frames.resolveRelative(f_bd_a, frame_a.R, frame_b.R);
annotation (
Documentation(info = "<html>\n<p>\nThis component consists of a <strong>universal</strong> joint at frame_a,\na <strong>spherical</strong> joint at frame_b and a <strong>prismatic</strong> joint along the\nline connecting the origin of frame_a and the origin of frame_b,\nsee the default animation in the following figure (the axes vectors\nare not part of the default animation):\n</p>\n\n<p>\n<img src=\"modelica://Modelica/Resources/Images/Mechanics/MultiBody/Joints/JointUPS.png\" alt=\"model Joints.Assemblies.JointUPS\">\n</p>\n\n<p>\nThis joint aggregation has no mass and no inertia and\nintroduces neither constraints nor potential state variables.\nIt is especially useful to build up more complicated force elements\nwhere the mass and/or inertia of the force element shall be taken\ninto account.\n</p>\n<p>\nThe universal joint is defined in the following way:\n</p>\n<ul>\n<li> The rotation <strong>axis</strong> of revolute joint <strong>1</strong> is along parameter\n vector n1_a which is fixed in frame_a.</li>\n<li> The rotation <strong>axis</strong> of revolute joint <strong>2</strong> is perpendicular to\n axis 1 and to the line connecting the universal and the spherical joint.</li>\n</ul>\n<p>\nThe definition of axis 2 of the universal joint is performed according\nto the most often occurring case. In a future release, axis 2 might\nbe explicitly definable via a parameter. However, the treatment is much more\ncomplicated and the number of operations is considerably higher,\nif axis 2 is not orthogonal to axis 1 and to the connecting rod.\n</p>\n<p>\nNote, there is a <strong>singularity</strong> when axis 1 and the connecting line are parallel\nto each other. Therefore, if possible n1_a should be selected in such a way that it\nis perpendicular to nAxis_ia in the initial configuration (i.e., the\ndistance to the singularity is as large as possible).\n</p>\n<p>\nAn additional <strong>frame_ia</strong> is present. It is <strong>fixed</strong> on the line\nconnecting the universal and the spherical joint at the\norigin of <strong>frame_a</strong>. The placement of frame_ia on this line\nis implicitly defined by the universal joint (frame_a and frame_ia coincide\nwhen the angles of the two revolute joints of the universal joint are zero)\nand by parameter vector <strong>nAxis_ia</strong>, an axis vector directed\nalong the line from the origin of frame_a to the spherical joint,\nresolved in frame_<strong>ia</strong>.\n</p>\n<p>\nAn additional <strong>frame_ib</strong> is present. It is <strong>fixed</strong> in the line\nconnecting the prismatic and the spherical joint at the\norigin of <strong>frame_b</strong>.\nIt is always parallel to <strong>frame_ia</strong>.\n</p>\n<p>\nNote, this joint aggregation can be used in cases where\nin reality a rod with spherical joints at each end are present.\nSuch a system has an additional degree of freedom to rotate\nthe rod along its axis. In practice this rotation is usually\nof no interest and is mathematically removed by replacing one\nof the spherical joints by a universal joint.\n</p>\n<p>\nThe easiest way to define the parameters of this joint is by moving the\nMultiBody system in a <strong>reference configuration</strong> where <strong>all frames</strong>\nof all components are <strong>parallel</strong> to each other (alternatively,\nat least frame_a, frame_ia and frame_ib of the JointUSP joint\nshould be parallel to each other when defining an instance of this\ncomponent).\n</p>\n</html>"),
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points = {
{30, 93},
{40, 90},
{30, 87},
{30, 93}},
lineColor = {128, 128, 128},
fillColor = {128, 128, 128},
fillPattern = FillPattern.Solid),
Line(
points = {
{-40, 90},
{40, 90}},
color = {128, 128, 128}),
Line(
points = {
{-25, 100},
{-10, 100}},
color = {0, 0, 255}),
Text(
extent = {
{-18, 90},
{19, 77}},
lineColor = {128, 128, 128},
textString = "s"),
Line(
points = {
{60, 90},
{40, 90},
{40, 98}},
color = {95, 95, 95},
thickness = 0.5),
Line(
points = {
{-40, 90},
{-40, 96},
{-40, 98}},
color = {135, 135, 135},
thickness = 0.5)}));
end JointUPS;